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Published for SISSA by SpringerReceived: October 22, 2012 Accepted: November 27, 2012 Published: December 20, 2012
Effective action for higher spin fields on (A)dS
backgrounds
Fiorenzo Bastianelli,a Roberto Bonezzi,a Olindo Corradinib,c and Emanuele Latinid,e
aDipartimento di Fisica, Universit`a di Bologna and INFN, Sezione di Bologna, via Irnerio 46, I-40126 Bologna, Italy
bCentro de Estudios en F´ısica y Matem´aticas Basicas y Aplicadas, Universidad Aut´onoma de Chiapas, Tuxtla Guti´errez 29050, Mexico cDipartimento di Fisica, Universit`a di Modena e Reggio Emilia,
Via Campi 213/A, I-41125 Modena, Italy
dInstitut f¨ur Mathematik, Universit¨at Z¨urich-Irchel, Winterthurerstrasse 190, CH-8057 Z¨urich, Switzerland eINFN, Laboratori Nazionali di Frascati,
CP 13, I-00044 Frascati, Italy
E-mail: bastianelli@bo.infn.it,bonezzi@bo.infn.it,
olindo.corradini@unach.mx,emanuele.latini@math.uzh.ch
Abstract: We study the one loop effective action for a class of higher spin fields by using
a first-quantized description. The latter is obtained by considering spinning particles,
characterized by an extended local supersymmetry on the worldline, that can propagate consistently on conformally flat spaces. The gauge fixing procedure for calculating the worldline path integral on a loop is delicate, as the gauge algebra contains nontrivial structure functions. Restricting the analysis on (A)dS backgrounds simplifies the gauge fixing procedure, and allows us to produce a useful representation of the one loop effective action. In particular, we extract the first few heat kernel coefficients for arbitrary even spacetime dimension D and for spin S identified by a curvature tensor with the symmetries of a rectangular Young tableau of D/2 rows and [S] columns.
Keywords: Extended Supersymmetry, Field Theories in Higher Dimensions, Sigma Mod-els
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Contents1 Introduction 1
2 Spinning particle on conformally flat spaces 3
2.1 Gauge-fixing in (A)dS 5
2.2 Regularization of supersymmetric nonlinear sigma models 8
3 Heat kernel expansion for higher spin fields in (A)dS 11
3.1 Integer spins 11 3.2 Half-integer spins 13 A Hamiltonian BRST quantization 15 B Propagators 18 C Modular integrals 19 C.1 Even N 19 C.2 Odd N 22 1 Introduction
Higher spin field theory is a topic that enters several aspects of modern theoretical physics. In this paper we quantize higher spin fields on (A)dS spaces using a worldline approach
and study their one loop effective action, extending the analysis of [1] that was restricted
to flat spacetimes.
The worldline approach to quantum field theory (see [2] for a review), has been known
to be an alternative tool to compute Feynman diagrams through the quantization of rel-ativistic point particles. More specifically, one loop effective actions in the presence of external fields find an efficient approach in terms of point particle path integrals computed on the circle, whereas field theory propagators are linked to particle path integrals on the
line. In particular, for relativistic higher spin fields (see [3–8] for reviews) the particle
approach might be particularly useful to extract information beyond the classical level. It is the main objective of the present manuscript to use a particle approach to compute the one loop effective action for higher spin fields in curved space. Indeed, extensions of the worldline approach to field theories with background gravity are feasible, as discussed for
example in [9–14].
The class of higher spin particles that we wish to treat here are those described by
the O(N ) spinning particles actions [15–18], that contain a fully-gauged extended
super-symmetry on the worldline. These models describe in first quantization higher spin fields
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in D = 4, and for spin S > 1 they live only in even space-time dimensions. In [22] the
conformal invariance was proven by showing that these particle models have classical back-ground reparametrization and Weyl invariance, thus leaving the conformal Killing vectors as generators of true symmetries. This result also implies that these models are consistent on generic conformally flat spaces. The particular coupling to (A)dS spaces was previously
known from the work of [23]. The class of higher spin fields treated here can be described
by higher spin curvature tensors that obey the symmetries of a Young tableau of D/2 rows
and [S] columns (see [24] for a discussion of the curvature tensors for half-integer spin).
More general types of higher spin fields could perhaps be described by using the detour
worldline methods of [25–27].
The gauge structure of our particle models on generic conformally flat spaces is quite
complex, as it contains non-trivial structure functions [22]. We find it simpler, for the
moment being, to investigate the one loop effective action on maximally symmetric spaces, i.e. (A)dS spaces, which allow for an algebraically simpler gauge fixing procedure. Weyl anomalies are generically present in quantum field theories, so that we expect to find a nontrivial effective action, as indeed we do.
One may also approach the problem directly in quantum field theory, as suitable actions
are known, see for example [28–36]. However we wish to suggest here the point of view that
many results are more efficiently obtained using first quantized methods.1 Recently the
heat kernel for some higher spin fields in (mostly) odd-dimensional maximally symmetric
spaces were computed using a group-theoretical approach [42–44]. Our approach deals with
a different set of multiplets on even-dimensional spaces. It would be useful to eventually compare the two approaches. Also, a different type of effective action with higher spin
backgrounds was studied in [45].
In subsequent sections we first present the gauge fixing of the models under study, then briefly review the regularization techniques needed to compute worldline path integrals in curved spaces. Finally we present the derivation of the worldline representation of the effective action. It is generically difficult to compute it in a closed form, so we aim here to calculate explicitly only the first few heat kernel coefficients for (A)dS backgrounds. For D > 2 these correspond to diverging terms that must be subtracted to renormalize the effective action. We perform the path integral computation with an arbitrary metric, as intermediate calculations might be useful for a larger class of backgrounds. Indeed, as mentioned above, these spinning particles are certainly consistent on conformally flat spaces. However, in that case the gauge fixing procedure is much more laborious and will not be attempted here.
The present analysis could be repeated step by step to carry out similar calculations
for the U(N ) spinning particle [46], which gives rise to higher spin fields living on complex
spaces [47] (treated already for the particular cases of N=1,2 on arbitrary Kahler manifolds
in [48,49]).
1
A worldline approach to quantum massive higher spins in (A)dS [37–41] can be treated along similar lines by dimensionally reducing the O(N) spinning particle used here.
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To conclude, the main results derived here are a worldline representation of the one
loop effective action for a class of higher spin fields on (A)dS spaces, see eq. (2.27), and
the calculations of the first few heat kernel coefficients, see eqs. (3.4), (3.5) for integer spin
and eqs. (3.11), (3.12) for half-integer spin.
2 Spinning particle on conformally flat spaces
The model we study here is the (fully) gauged counterpart of the mechanical model with action S = Z 1 0 dtpµx˙µ+ i 2ψ a iψ˙ia− 1 2pµp µ, i = 1, . . . , N (2.1)
where a set of global worldline symmetries (time translation, N supersymmetries, O(N )
R-symmetry) are rendered local to guarantee unitarity. Here xµ and pµ are spacetime
coordinates and momenta of the particle, whereas ψa
i are Majorana fermions, with a a
flat Lorentz index. The resulting phase-space action identifies the so-called O(N ) spinning particle model and, when considering a curved target space, reads
S[x, p, ψ, E; g] = Z 1 0 dt " pµx˙µ+ i 2ψ a iψ˙ia− eH − iχiπµeµaψai | {z } Qi −1 2aijiψ| i{z· ψ}j Jij # (2.2)
with H = H0− 18Rabcd ψa· ψbψc· ψd and H0 = 12gµνπµπν being the kinetic hamiltonian
written in terms of the covariant momenta πµ= pµ−2iωµabψiaψib. From (2.2) one recognizes
the supercharges Qi and the O(N ) symmetry generators Jij. E collectively denotes the
worldline gauge fields E = (e, χi, aij), i.e. einbein, gravitini and O(N ) gauge fields
respec-tively. This model describes the first quantization of a particular mixed-symmetry higher spin particle in D = 2d even-dimensional curved space, that generically (for N > 2) must
be conformally flat. The spectrum of the model for N > 2 is empty in odd dimensions [18].
For even N = 2n the model describes equations of motion (the Dirac constraints) for a bosonic field strength characterized by a rectangular Young tableau with n columns and d rows. For odd N = 2n + 1 the model describes equations of motion for a fermionic field strength, a spinor-tensor with a tensor structure characterized by the same n × d Young
tableau. For D = 4 this involves all possible massless representations of the Poincar´e
group, that at the level of gauge potentials are given by totally symmetric (spinor-)
ten-sors, whereas for D > 4 it corresponds to conformal multiplets only [19–21]. The euclidean
configuration space action, that one obtains after integrating out the momenta pµ and
Wick rotating, reads S[y, E; g] = Z 1 0 dτ " 1 2egµν ˙ xµ− χiψiµx˙ν − χjψνj (2.3) +1 2ψ a i δijδab∂τ+ ˙xµωµabδij − aijδab ψjb−e 8Rabcd ψ a· ψbψc· ψd #
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with y = (xµ, ψia) being the “matter” fields. For arbitrary N and generic curved
back-grounds the gauge symmetry generators (H, Qi, Jij) do not form a first class algebra.
How-ever in [22] it was found that, if the background is conformally flat, they form a (nonlinear)
first-class constraint algebra and the previous action is gauge-invariant under the
transfor-mations induced by the gauge symmetry generator G = ξH + iiQi+12αijJij := ΞAGA.2
At the quantum level the constraint algebra on conformally flat spaces closes as well, provided one adds to the hamiltonian an improvement term proportional to the scalar of curvature, namely H = H0− 1 8Rabcd ψ a· ψbψc· ψd−(N − 2)(D + N − 2) 16(D − 1) R (2.4)
with the kinetic operator given by
H0 = 1 2 πa− iωbbaπa πa = eµaπµ, πµ= g1/4pµg−1/4− i 2ωµabψ a iψbi . (2.5)
Here we use a path integral formalism and find it more convenient to use the (euclidean) configuration space action
S[y, E; g] = Z 1 0 dτ " 1 2egµν ˙ xµ− χiψiµ ˙ xν − χjψjν +1 2ψ a i δijδab∂τ+ ˙xµωµabδij − aijδab ψjb−e 8Rabcd ψ a· ψbψc· ψd −e(N − 2)(D + N − 2) 16(D − 1) R # (2.6)
that is (2.3) with the addition of the improvement term. The associated path integral
evaluated on the circle S1
Γ[g] = Z S1 DE Dy Vol (Gauge) e −S[y,E;g] (2.7)
gives a representation of the one loop effective action for the aforementioned higher spin field coupled to external gravity. It is defined by taking bosonic fields with periodic boundary conditions and fermionic fields with antiperiodic boundary conditions.
In order to be able to perform computations two preliminary issues have to be taken care of:
i) Firstly, the worldline action must be suitably gauge-fixed; i.e. the gauge fields E must be fixed to some specific configuration that will depend upon a set of modular parameters that must be integrated over. In the present case the gauge symmetry algebra, associated to the above generators, is nonlinear, i.e. commutators of pairs of generators involve structure functions and not structure constants. Therefore one must use more powerful hamiltonian BRST methods to gauge fix the action in its hamiltonian form.
2
For N 6 2 the R-symmetry group is either trivial or abelian, and the algebra closes on an arbitrary background.
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ii) The resulting gauge-fixed action depends only upon “matter” fields and modular parameters. However, in curved space, it still is a nonlinear sigma model, so that for perturbative computations one usually Taylor expands the metric about a fixed point of the circle. This results in an infinite set of vertices. In addition some Feynman diagrams present ambiguities and need to be regularized. This is a well-known fact, and several regularization schemes have been used in the past to compute such path
integrals; see [60] for an overall review. Up to recently only quantum-mechanical
path integrals in curved space with N 6 2 had been used: these path integrals, in the worldline formalism, correspond to the first quantization of spin S 6 1 fields
in curved space. More recently in [61] the regularization of nonlinear sigma models
with arbitrary N was considered, having in mind applications to the O(N ) spinning
particles. What studied in [61] are the globally supersymmetric counterparts of the
models studied here. That is enough for the present purposes as the gauging does not introduce additional ambiguities.
2.1 Gauge-fixing in (A)dS
In this section we describe the gauge-fixing of the O(N ) spinning particle propagating on (A)dS spaces. For such backgrounds the Riemann curvature can be written as
Rabcd = b(ηacηbd− ηadηbc) (2.8)
where Λ = (D − 1)(D − 2)b is the cosmological constant. Let us start considering the action
in hamiltonian form. At the classical level (cfr. (2.2)), in (A)dS spaces the hamiltonian
constraint reduces to H = H0−4bJijJij and the first-class algebra reduces to a quadratic
algebra (curly brackets here are graded Poisson brackets)
{Qi, Qj} = −2iδijH + ib JikJjk− 1 2δijJklJkl {Jij, Jkl} = δjkJil− δikJjl− δjlJik+ δilJjk {Jij, Qk} = δjkQi− δikQj, {H, Qi} = {H, Jij} = 0 (2.9)
that can be used to obtain the corresponding transformations of the gauge fields.
Upon canonical quantization the latter quadratic algebra turns into the following (anti-)commutation relations
{Qi, Qj} = 2δijH −
b
2(JikJjk+ JjkJik− δijJklJkl)
[Jij, Jkl] = iδjkJil− iδikJjl− iδjlJik+ iδilJjk
[Jij, Qk] = iδjkQi− iδikQj, [H, Qi] = [H, Jij] = 0 (2.10)
with the hamiltonian constraint given by (2.4), that in (A)dS reduces to
H = H0−
b
4JijJij − bA(D) (2.11)
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In order to gauge fix the locally symmetric O(N ) spinning particle action (with
quan-tum gauge algebra given in (2.10)) we use the hamiltonian BRST method reviewed in
appendix A. Basically, we define ghost fields CA = (C, Ci, Cij) and ghost momenta PA =
(P, Pi, Pij) for all constraint generators GA= (H, Qi, Jij), such that [PA, CB} = −iδAB and
write the quantum BRST operator as a graded sum Ω =P
p≥0
(p)
Ω . Starting from
(0)
Ω = CAGA= CH + CiQi+ CijJij (2.12)
and imposing the nilpotency of the BRST charge, we can recursively obtain higher antighost-number operators. Setting
[GA, GB} = FABC (z) GC (2.13)
with zα= (pµ, xµ, ψia) and FABC (z) structure functions, for the algebra (2.10) we get
(1) Ω = i 2(−) εACACBFC BAPC = −iCiCiP − 2CkCkiPi+ 2CikCkjPij− i b 4 CiCiJklPkl− 2CiCjJikPjk (2.14) and (3) Ω = b 2 24 CiCjCkClPijPkmPlm− 3CmCmCiCjPikPjlPkl+ CkCkClClT r(Pij3) (2.15) (2) Ω = (p) Ω = 0 , p > 3 . (2.16)
One can thus write the quantum gauge-fixed hamiltonian operator as
Hqu = HBRST− i{K, Ω}
where the first term is a BRST-invariant hamiltonian and K a gauge fixing fermion: the latter is BRST-invariant for any choice of K thanks to the nilpotency of Ω. In the present
case since H itself enters as a constraint we can set HBRST = 0 and thus have
Hqu = −i{K, Ω} . (2.17)
Let us now use the gauge-fixing fermion
K = − ˆEAPA, EˆA= β, 0,θij 2 (2.18)
with θij a N × N skew diagonal matrix, dependent on [S] = [N/2] := n angular variables
θk, with k = 1, . . . , n. Here S = N/2 is the “spin” of the particle. With this choice one
obtains the hamiltonian operator
Hqu = βH +
1
2θijJij− θijCiPj− 2θijCimPjm (2.19)
and consequently the gauge-fixed path integral can be written as
Γ[g] = KN Z ∞ 0 dβ β n Y k=1 Z 2π 0 dθk 2π Z S1 Dz DC DP eiSqu[z,C,P, ˆE;g] (2.20)
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with phase space action
Squ[z, C, P, ˆE; g] = Z 1 0 dt h pµx˙µ+ i 2ψ a iψ˙ia+ ˙CAPA− Hqu i (2.21) Hqu = β 1 2gµν(x)π µπν −b 4JijJij− bA(D) +1 2θijJij− θijCiPj − 2θijCimPjm (2.22) and πµ = pµ− i 2ω µ
abψaiψib. Above KN is a normalization factor that implements the
reduction to a fundamental region of moduli space
KN = ( 2 2nn!, N = 2n 1 2nn!, N = 2n + 1 (2.23)
as discussed in [1]. Integrating out particle momenta leads to a configuration space path
integral that involves the action
Squ[y, C, P, ˆE; g] = Z 1 0 dt " 1 2βgµνx˙ µx˙ν + i 2ψaiDtψ a i + β b 4JijJij+ bA(D) −1 2θijJij −P ˙C + Pi(δij∂t− θij)Cj − Pij(δimδjp∂t− θimδjp+ θjmδip)Cmp # (2.24)
where Dtψai = ˙ψia+ ˙xµωµabψib. A Wick rotation to euclidean time yields
Γ[g] = KN Z ∞ 0 dβ β n Y k=1 Z 2π 0 dθk 2π Z S1 Dy DC DP e−Squ[y,C,P, ˆE;g] (2.25)
with the euclidean version of the action given by
Squ[y, C, P, ˆE; g] = 1 β Z 1 0 dτ " 1 2gµνx˙ µx˙ν +1 2ψai δijDτ− θij ψaj − b 4JijJij− β 2bA(D) −P ˙C + Pi(δij∂t− θij)Cj+ Pij(δimδjp∂t− θimδjp+ θjmδip)Cmp # . (2.26)
where we have Wick-rotated the O(N ) fields θij → iθij and the ghost momenta PA→ iPA.
Here Dτ is represented by the same covariant derivative as given above, with “dot” now
representing derivative with respect to τ . Fermions and ghosts have been suitably rescaled
in order to have a common β1 in front of the action. In the following we perturbatively
compute the above path integral. Although the latter is defined on (A)dS spaces, for convenience we keep the geometry arbitrary and only at the end do we fix it to (A)dS. In
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above action. Integrating over the ghost fields yields
Γ[g] = KN Z ∞ 0 dβ β n Y k=1 Z 2π 0 dθk 2π
Det(∂τ− θvec)ABC
−1 Det0(∂τ− θadj)P BC Z S1 DxDψ exp −1 β Z 1 0 dτ 1 2gµνx˙ µx˙ν +1 2ψai δijDτ− θij ψja −1 8Rabcdψ a· ψbψc· ψd− β2(N − 2)(D + N − 2) 16(D − 1) R ! (2.27)
where θvec and θadj denote the gauge-fixed O(N ) potentials in the vector and adjoint
representation, respectively. Det0 indicates a determinant without its zero modes, and Dx
is the reparametrization invariant measure. Below we consider a short-time perturbative approach to the above nonlinear sigma model path integral.
2.2 Regularization of supersymmetric nonlinear sigma models
For a particle in curved space, the passage between the operatorial representation of the transition amplitude and its path integral counterpart is in general not straightforward, as the latter involves a nonlinear sigma model that perturbatively gives rise to superficial di-vergences. These divergences are compensated by vertices arising from the nontrivial path integral measure, but finite ambiguities remain that need to be dealt with by specifying a regularization scheme. This is well studied for models with global (super)symmetries
(see [60] for a review). However it is clear that gauging does not introduce further
diver-gences. Indeed upon gauge fixing, the gauged model reduces essentially to the ungauged one. Moreover the ghosts do not couple to the target space geometry and just produce the correct measure for integration over the moduli space.
In [61] we considered the regularization of the spinning particle model with hamiltonian
H = H0+ αRabcdψiaψibψjcψdj + V (2.28)
with H0 given by (2.5). The corresponding euclidean classical action in configuration space
is given by S = 1 β Z 1 0 dτ " 1 2gµνx˙ µx˙ν+ 1 2ψaiDτψ a i + αRabcdψaiψibψcjψjd+ β2V # (2.29)
and, for α = −18, is nothing but the ungauged version of the nonlinear sigma model of
the previous section. We found that such path integral reproduces the transition
ampli-tudes that satisfies the Schr¨odinger equation with hamiltonian (2.28) provided we add the
counterterm VCT = − 1 8+ αN 2 R + 1 8g µνΓρ µλΓ λ νρ+ N16ωµabω µab, T S − 18+αN2 R −241(Γµλρ )2+ N24ωµabωµab, M R − 18+αN2 R , DR (2.30)
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Since the process of gauging does not introduce further ambiguities than those already
taken into account in [61], we conclude that the regularization there discussed is suitable
for the model of the previous section, provided one sets α = −18 and
V = VCT −
(N − 2)(D + N − 2)
16(D − 1) R . (2.31)
Above T S refers to Time Slicing regularization [50, 51], M R refers to Mode
Regulariza-tion [52–56] and DR refers to Dimensional Regularization [10,12,57–59], that are the three
regularization schemes developed in the past to treat one-dimensional nonlinear sigma mod-els (particles in curved space). In the present work we adopt DR to compute the short
time perturbative expansion of (2.27). We parametrize the coordinates of the circle as
xµ(τ ) = xµ+ qµ(τ ), where xµ is the initial/final point of the circle and qµ(τ ) are
quan-tum fluctuations with Dirichlet boundary conditions qµ(0) = qµ(1) = 0. Fermions have
antiperiodic boundary conditions on the circle and have no zero modes. We then expand
the metric and the spin connection about the point xµusing Riemann normal coordinates,
and get gµν(x(τ )) = gµν+ 1 3Rαµνβq αqβ+ 1 6∇γRαµνβq αqβqγ + Rαβµνγδqαqβqγqδ+ O(q5) (2.32) ωµab(x(τ )) = 1 2Rαµabq α+1 3∇αRβµabq αqβ+ 1 8∇α∇βRγµabq αqβqγ + 1 30∇α∇β∇γRδµabq αqβqγqδ+ O(q5) (2.33)
where Rαβµνγδ = 201 ∇δ∇γRαµνβ+452 RαµσβRγσνδ. All the tensors are here evaluated at the
initial point xµ. Above we only give the terms that are needed to obtain a perturbative
expansion to order β2. We thus get
Γ[g] = KN Z dDx Z ∞ 0 dβ β n Y k=1 Z 2π 0 dθk 2π
Det(∂τ− θvec)ABC
−1 Det0(∂τ − θadj)P BC × Z DBC DqDaDbDcD ¯ψDψDη e−β1 R1 0 dτ 1 2gµν( ˙qµq˙ν+aµaν+bµcν)+ P kψ¯ak(∂τ+iθk)ψak+ 1 2ηa∂τηa × e−Sint (2.34)
where we have exponentiated the reparametrization invariant measure by means of measure
ghosts a, b, c [52,53] and have complexified the 2n fermions ψ; the leftover uncomplexified
Majorana fermion η is only present when the number of supersymmetries N is odd — i.e. for half-integer spin. From the quadratic part of the action one gets the path integral
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higher order terms form the interacting action
Sint = 1 β Z 1 0 dτ " 1 6Rαµνβq αqβ+ 1 12∇γRαµνβq αqβqγ+1 2Rαβµνγδq αqβqγqδ ( ˙qµq˙ν+ aµaν + bµcν) + 1 2Rαµabq α+1 3∇αRβµabq αqβ+1 8∇α∇βRγµabq αqβqγ + 1 30∇α∇β∇γRδµabq αqβqγqδ ˙ qµ n X k=1 ¯ ψkaψbk+1 2η aηb ! + α Rabcd+ qα∇αRabcd+ 1 2q αqβ∇ α∇βRabcd ψa· ¯ψb ψc· ¯ψd+ ηcηd + β2 V + qα∇αV + 1 2q αqβ∇ α∇βV # (2.35) whose path integral average is computed using the Wick theorem. We thus get
Γ[g] = Z ∞ 0 dβ β Z dDxp|g| (2πβ)D/2 n Y k=1 Z 2π 0 dθk 2πd(θ; D, S) D e−Sint E (2.36) with √ |g|
(2πβ)D/2 being the normalization of the bosonic path integral in D dimensions with
Dirichlet boundary conditions, whereas the fermionic normalization contributes to the mod-uli integrand
d(θ; D, N ) = KN
Det(∂τ− θvec)ABC
D2−1 Det0(∂τ− θadj)P BC (2.37) = 2 2nn! n Y k=1 2 cosθk 2 D−2Y k<l 2 cosθl 2 2 −2 cosθk 2 22 , N = 2n 2D2−1 2nn! n Y k=1 2 cosθk 2 D−2 2 sinθk 2 2Y k<l 2 cosθl 2 2 −2 cosθk 2 22 , N = 2n + 1 that integrated gives
Dof (D, N ) = n Y k=1 Z 2π 0 dθk 2π d(θ; D, N ) := a0, (2.38)
the number of degrees of freedom for the higher spin field described by the locally
supersym-metric spinning particle model with N supersymmetries [1], i.e. the physical polarizations
of a particle of spin S = N/2. By factoring out the number of degrees of freedom, we can finally write the above effective action in a compact way as
Γ[g] = a0 Z ∞ 0 dβ β Z dDxp|g| (2πβ)D/2 DD e−Sint EE = Z ∞ 0 dβ β Z(β) (2.39)
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with
· · · representing the average over the path integral and over the moduli space. Hence, for the effective action density in proper time we get
Z(β) = a0 Z dDxp|g| (2πβ)D/2 DD e−Sint EE = Z dDxp|g| (2πβ)D/2 a0+ a1β + a2β2+ O(β3) (2.40)
and we parametrize the Seleey-DeWitt coefficients ai as follows
a0
1 + v2Rβ + (v3R2abcd+ v4R2ab+ v5R2+ v6∇2R)β2+ O(β3)
. (2.41)
Next we compute the numerical coefficients vi.
3 Heat kernel expansion for higher spin fields in (A)dS
Equipped with the results of the previous sections we can now compute the heat kernel in a perturbative expansion for higher spin fields on (A)dS spaces, using the O(N ) spinning particle representation discussed above. Although in the previous sections we gauge fixed the locally supersymmetric action for maximally symmetric spaces only, here we compute the expansion keeping an unspecified metric in the sigma model and only at the end of the section will we specialize to (A)dS spaces. This we do mostly for future convenience, as intermediate results might be useful when considering more general spacetimes, such as the conformally flat spaces. Since in the following we adopt dimensional regularization, the total potential acquires the form:
V = w R , with w(D, N, α) := wCT(N, α) + w(A)dS(D, N ) where wCT(N, α) = − 1 8+ αN 2 , w(A)dS(D, N ) = −(N − 2)(D + N − 2) 16(D − 1) , (3.1) as follows from (2.30), (2.31). 3.1 Integer spins
For this case we set N = 2n. One can complexify fermions and, with the help of propagators
given in appendix B, one gets for the perturbative average
D e−Sint E = exp ( −β " 1 24 + α n − X k cos−2θk 2 ! + w # R + β2 " − 1 720R 2 αβ + 1 720− 1 192 X k cos−2 θk 2 ! R2αµνβ− 1 480+ w 12 ∇2R # −αβ 2 12 n − X k cos−2 θk 2 ! ∇2R + (αβ)2 " X k cos−2 θk 2 !2 −1 2 X k cos−4θk 2 ! R2αµνβ + 2 X k cos−2 θk 2 − X k cos−4 θk 2 ! R2αβ # + O(β3) ) , (3.2)
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that, for α = −1/8 reduces to D e−Sint E = 1 − β 1 − 3n 24 + 1 8 X k cos−2 θk 2 + w ! R + β2 ( 1 2 1 − 3n 24 + 1 8 X k cos−2θk 2 + w !2 R2 + − 1 720− 1 32 X k cos−4θk 2 + 1 32 X k cos−2 θk 2 ! R2ab + 1 720− 1 192 X k cos−2θk 2 + 1 64 X k cos−2 θk 2 !2 − 1 128 X k cos−4 θk 2 ! R2abcd− 1 − 5n 480 + 1 96 X k cos−2 θk 2 + w 12 ! ∇2R ) + O(β3) , (3.3) with w = w(D = 2d, N = 2n, α =−1 8) = − (N − 2)(N − 1) 16(D − 1) = − (2n − 1)(n − 1) 8(2d − 1) .
We are ready now to extract the Seeley-DeWitt coefficients for arbitrary integer spin S = n
in arbitrary even dimension D = 2d; to this aim we integrate (3.3) against the modular
measure given in (2.36), (2.37) and get:
a0 = 1 , n = 0 2n−1 (2d − 2)! [(d − 1)!]2 n−1 Y k=1 k(2k − 1)!(2k + 2d − 3)! (2k + d − 2)!(2k + d − 1)!, n > 0 (3.4) and v2 = 3n − 1 24 − 1 8I1− w v3 = 1 720− n(n + 1) 256 + 3n + 1 384 I1+ 3 256I2+ 1 256I3 v4 = − 1 720+ n(n + 1) 64 − n 32I1+ 1 64I2− 1 64I3 v5 = 1 2 9n2− 21n + 2 1152 − w(3n − 1) 12 + w 2 +1 2 5 − 3n 192 + w 4 I1+ 1 256 I2+ I3 v6 = 5n − 1 480 − w 12 − 1 96I1 (3.5) with I1 = 2n(n + d − 2) 2d − 3 I2 = 4n(n − 1)(n + d − 1)(n + d − 2) (2d − 3)(2d − 1) I3 = n(n + 1)(4n2− 1) (2d − 3)(2d − 5) (3.6)
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Detailed computation of modular integrals is given in appendix C. Let us now briefly
comment on the results described above in (3.4), (3.5):
• For n = 0, the formalism describes a conformally coupled scalar field and the expected results are easily obtained.
• For n = 1, (3.4), (3.5) reproduce the well known Seeley-DeWitt coefficients for a
degree (d − 1) differential form (vector field in D = 4) [12] on a general background.
• For n ≥ 2, the spinning particle consistently propagates on conformally flat manifolds. However, for this case, in the previous sections we limited the computation of the BRST charge to (A)dS spaces. Hence the structure of the Seeley-DeWitt coefficients reduces to a0 1 + v2Rβ + vR2β2 with v = 1 d(2d − 1)v3+ 1 2dv4+ v5. Example: D = 4 , spin n
In 4-dimensional space-time the model describes completely symmetric tensors of spin n, and the Seeley-DeWitt coefficients are given by:
a0 = ( 1 , n = 0 2 , n > 0 , v2 = − n2 6 , v3 = 1 720− n2 96 v4 = − 1 720− n2 48 + n4 12, v5 = 1 96n 2− 1 36n 4, v 6 = 1 720− 1 72n 2 , (3.7)
When n ≥ 2 the restriction to (A)dS yields:
v = 1 6v3+ 1 4v4+ v5= − 1 8640+ 1 288n 2− 1 144n 4 (3.8)
We again recognize for n = 0, 1 the known coefficients for a conformally improved scalar
and an ordinary spin one vector field. For n > 0 the first coefficient a0 represents the two
polarizations of massless particles of spin n.
The case of n = 2 corresponds to a linearized graviton on a fixed background, but this is true only in D = 4. In other dimensions one has a different field content compatible with conformal invariance.
3.2 Half-integer spins
In such a case one can only complexify 2n fermions. The left-over one has no θ, and one thus gets D e−SintE= exp ( −β " 1 24 + w + α n − X k cos−2θk 2 !# R + β2 " − 1 720R 2 αβ− 7 5760+ 1 192 X k cos−2 θk 2 ! R2αµνβ− 1 480+ w 12 ∇2R # −αβ 2 12 n − X k cos−2 θk 2 ! ∇2R
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+ (αβ)2 " X k cos−2 θk 2 !2 +X k cos−2 θk 2 − 1 2 X k cos−4θk 2 ! R2αµνβ + 2 X k cos−2 θk 2 − X k cos−4θk 2 ! R2αβ # + O(β3) ) , (3.9)that, for α = −1/8, reduces to D e−Sint E = 1 − β 1 − 3n 24 + 1 8 X k cos−2 θk 2 + w ! R + β2 ( 1 2 1 − 3n 24 + 1 8 X k cos−2θk 2 + w !2 R2 + − 1 720− 1 32 X k cos−4θk 2 + 1 32 X k cos−2 θk 2 ! R2ab + − 7 5760+ 1 96 X k cos−2 θk 2 + 1 64 X k cos−2θk 2 !2 − 1 128 X k cos−4 θk 2 ! R2abcd− 1 − 5n 480 + w 12 + 1 96 X k cos−2θk 2 ! ∇2R ) + O(β3) (3.10)
where now we use
w = w(D = 2d, N = 2n + 1, α =−18) = −(N − 2)(N − 1)
16(D − 1) = −
n(2n − 1)
8(2d − 1) .
We compute, in analogy with the previous section, the Seeley-DeWitt coefficients for
ar-bitrary half-integer spin S = n +12 in arbitrary even dimension 2d, represented by
spinor-tensors corresponding to potentials with rectangular Young tableaux of n columns and d − 1 rows; we get: a0 = 2d−2+n d (2d − 2)! [(d − 1)!]2 n−1 Y k=1 (k + d − 1)(2k + 1)!(2k + 2d − 3)! (2k + d − 1)!(2k + d)! (3.11) and v2 = 3n − 1 24 − 1 8eI1− w v3 = − 7 5760− n(n + 1) 256 + 3n + 7 384 eI1+ 3 256eI2+ 1 256eI3 v4 = − 1 720+ n(n + 1) 64 − n 32eI1+ 1 64eI2− 1 64eI3 v5 = 1 2 9n2− 21n + 2 1152 − w(3n − 1) 12 + w 2 +1 2 5 − 3n 192 + w 4 eI1+ 1 256 eI2+ eI3 v6 = 5n − 1 480 − w 12 − 1 96eI1 (3.12)
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with eI1 = 2n(n + d − 1) 2d − 3 eI2 = 4n(n − 1)(n + d − 1)(n + d) (2d − 3)(2d − 1) eI3 = n(n + 1)(2n + 1)(2n + 3) (2d − 3)(2d − 5) . (3.13)The modular integrals are again computed in details in appendixC.
In the half-integer spin case the spinning particle model we start with is consistent on
any background only if n = 0 (i.e. spin 12). When n ≥ 1 we restrict our analysis to (A)dS
spaces and at order β2 in the expansion of the effective action the only term that survives
is a0vR2 where, again, v = d(2d−1)1 v3+2d1 v4+ v5.
Example: D = 4 , spin n + 12
In 4-dimensional space-time we describe spinor-tensors with n completely symmetric
vector indices and one spinor index i.e. spin n + 12. The Seeley-DeWitt coefficients we
find are: a0= 2 , v2= − (2n + 1)2 24 , v3 = − 7 5760− n 96 − n2 96 v4= − 1 720+ n 48 + 5n2 48 + n3 6 + n4 12, v5 = 1 1152+ 1 144n − 5 96n 2− 19 288n 3− 1 144n 4, v6= − 1 480− 1 72n − 1 72n 2. (3.14)
When n = 0 the previous formulas reproduce the well know Seeley-DeWitt coefficients for
a spinor field [10], while for n ≥ 1 in (A)dS we get:
v = 11 34560+ n 96 − n2 36 − 7n3 288+ n4 72.
Let us stress again that in 4 dimension we recognize in a0 = 2 the two polarizations of a
massless half-integer spin field.
Acknowledgments
The work of FB was supported in part by the MIUR-PRIN contract 2009-KHZKRX. The work of OC was partly funded by SEP-PROMEP/103.5/11/6653. EL acknowledges partial support of SNF Grant No. 200020-131813/1. OC and EL are grateful to the Dipartimento di Fisica and INFN Bologna for hospitality and support while parts of this work were completed.
A Hamiltonian BRST quantization
The hamiltonian BRST formalism is a construction that allows to convert the local (gauge) symmetry of the unfixed action (in hamiltonian form) to a global symmetry of the gauge-fixed action. It makes use of the double aspect that first-class generators have, as
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One defines a differential δ (the Koszul-Tate differential) that acts as a derivative in
the directions orthogonal to the constrained phase-space manifold and is nilpotent, δ2 = 0.
Hence the definition
δzα= 0, zα = (pµ, xµ, ψia) . (A.1)
Moreover, one extends the phase space defining ghosts CA and ghost momenta PA, such
that {PA, CB} = −δAB and
δCA= 0, δPA= −GA (A.2)
with GAfirst class constraints. The operator δ thus defines a natural grading, characterized
by the antighost number
gh(δ) = −1, gh(z) = 0 = gh(C), gh(P) = 1 . (A.3)
Note that the bracket itself in the ghost sector has antighost number −1. Another grading is the Grassmann parity
εA:= ε(GA) (A.4)
so that, since ε(δ) = 1, we have
ε(CA) = ε(PA) = εA+ 1, mod 2 . (A.5)
One also introduces another derivative d that acts parallel to the gauge orbits. It is defined on functions of the original phase space, φ(z), as
dφ = {φ, CAGA} = {φ, GA}CA, gh(d) = 0, ε(d) = 1 . (A.6)
Finally one seeks a differential s that is a graded sum of δ, d and higher order (in antighost number) derivatives, such that it results nilpotent on the extended phase space involving ghosts
s = δ + d + “higher order terms“ , s2= 0 . (A.7)
Thanks to antighost grading, nilpotency of s implies
δ2 = 0 (A.8)
dδ + δd = 0 (A.9)
d2 = −{δ, ∆} (A.10)
· · ·
Equations (A.9), (A.10) mean that d is a “differential modulo δ”. The first one is satisfied,
along with the grading properties, if one defines the following rules for the action of d on the extended phase space
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where F ’s are structure functions and only depend upon the original phase space variables
GA, GB = FABC GC, FABC = FABC (z) . (A.12)
One then seeks a BRST operator Ω
Ω =X p≥0 (p) Ω , gh (p) Ω = p (A.13)
that implements the action of the differential s as
sΦ = {Φ, Ω} (A.14)
with Φ(z, C, P) a function of the extended phase space variables, where
(0) Ω = CAGA (A.15) so that δΦ = n Φ, (0) Ω o C P = {Φ, CA}GA (A.16)
with the lowerscript CP meaning that the bracket is only taken in the ghost sector. It is
trivial to check that (A.16) works correctly on the extended phase space variables. For a
function of the original phase space we obviously have dφ(z) = n φ, (0) Ω o orig = {φ, GA}CA.
Finally, thanks to the Jacobi identity, the nilpotency conditions turns into n
Ω, Ω o
= 0 . (A.17)
Higher order operators have the form
(p)
Ω = CB1· · · CBp+1UA1···Ap
B1···Bp+1PA1· · · PAp, U = U (z) (A.18)
so that the nilpotency equation (A.17), with the help of (A.16), allows to write
δ(0)Ω = 0 (A.19) δ (p+1) Ω +1 2 p X k=0 n(p−k) Ω , (k) Ω o orig+ p−1 X k=0 n(p−k) Ω , (k+1) Ω o C P ! = 0 , p ≥ 0 . (A.20)
For example, it is easy to find the next-to-leading operator
(1) Ω as δ (1) Ω = −1 2 n(0) Ω , (0) Ω o orig = 1 2(−) εACACBFC BAGC = δ −1 2(−) εACACBFC BAPC (A.21) so that, modulo a δ-exact term
(1)
Ω = −1
2(−)
εACACBFC
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One thus recursively fixes all other terms in the graded expansion. If the constraint algebra is linear (i.e. it is a Lie algebra) the expansion stops at p = 1.
The gauge fixed action in hamiltonian form reads
Sgf = Z 1 0 dt ˙ zαaα+ ˙CAPA− HBRST − n K, Qo (A.23)
where aα are momenta conjugated to the phase space variables z (indices are contracted
with the symplectic identity matrix), HBRST is the BRST-invariant extension of the
ex-tended hamiltonian and K an arbitrary gauge-fixing fermion. This action is BRST invariant for any K. An important example concerns algebraic gauges for which the gauge fields are
fixed to ˆEA: in such a special case
K = − ˆEAPA (A.24)
for which
n
K, Qo= ˆEAGA− (−)εAEˆACBFBAC PC+ · · · . (A.25)
The above technique to construct the BRST charge Ω is known as Koszul-Tate resolution. Below we use the Koszul-Tate resolution to study an interesting class of non lie rank 3 superalgebra, and to construct the gauge fixed action for O(N ) spinning particles prop-agating on (A)dS target spaces. We do it directly at the quantum level where Poisson
brackets are replaced by (anti-)commutators, such as [PA, CB} = −iδAB and PA are taken
to be (anti-)hermitian when (anti-)commuting whereas CA are always hermitians. The
master formula (A.17) is now a nilpotency condition on the BRST charge, Ω2 = 0. We
thus have (0) Ω = CAGA, (1) Ω = i 2(−) εACACBFC BAPC, . . . . (A.26)
The hamiltonian operator is given by Hqu = HBRST − i{K, Ω}, with HBRST a
BRST-invariant hamiltonian and K a gauge-fixing fermion.
B Propagators
Propagators are obtained by inverting the differential operators appearing in the quadratic
action β1R1 0 dτ 1 2gµν( ˙q µq˙ν+ aµaν+ bµcν) +P kψ¯ak(∂τ + iθk)ψka+12ηa∂τη a qµ(τ )qσ(σ) = −βgµν∆(τ, σ) (B.1) aµ(τ )aσ(σ) = βgµν∆ gh(τ, σ) (B.2) bµ(τ )cσ(σ) = −2βgµν∆ gh(τ, σ) (B.3) ψa k(τ ) ¯ψbk0(σ) = βδkk0δab∆AF(τ − σ, θk) (B.4) ηa(τ )ηb(σ) = βδab∆ AF(τ − σ, 0) (B.5) with ∆(τ, σ) = (τ − 1)σθ(τ − σ) + (σ − 1)τ θ(σ − τ ) (B.6) ∆gh(τ, σ) =••∆(τ, σ) = δ(τ, σ) (B.7)
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and ∂τ + iθk ∆AF(τ − σ, θk) = δA(τ − σ) (B.8) that yields ∆AF(τ − σ, θk) = e−iθk(τ −σ) 2 cosθk 2 eiθk/2θ(τ − σ) − e−iθk/2θ(σ − τ ) . (B.9) Hence ∆AF(0, θk) = i 2tan θk 2 (B.10) ∆AF(τ − σ, 0) = 1 2(τ − σ) . (B.11) C Modular integralsIn this appendix we are going to show the detailed calculation of the modular integrals
required to find the Seleey-DeWitt (SDW) coefficients presented in section 3. We will
always consider even dimensional spacetime with D = 2d, and we shall distinguish the two cases of even and odd N , although the techniques will be the same.
C.1 Even N
We compute the modular integrals for the even N = 2n case. First of all, we define the
modular average of an arbitrary function f (θj) of the moduli θj; by using the measure
given in (2.36) and (2.37), and taking into account that modular integrals are even under
θi→ 2π − θi, we have: hhf (θj)iiE := 1 a0 2 n! n Y i=1 Z π 0 dθi 2π 2 cosθi 2 D−2 Y k<l " 2 cosθk 2 2 − 2 cosθl 2 2#2 f (θj) (C.1)
where a0 is the normalization factor giving the degrees of freedom, that ensures hh1iiE = 1,
and reads a0 := 2 n! n Y i=1 Z π 0 dθi 2π 2 cosθi 2 D−2 Y k<l " 2 cosθk 2 2 − 2 cosθl 2 2#2 . (C.2)
The result for (C.2) is already known from [1], but will be rederived here. Since all the
integrals we need will be expressed as generalizations of the Selberg’s integral, it is
conve-nient to change variables as xi = sin2 θ2i, ranging from 0 to 1. The average of a function
f (xj) := f (θ(xj)) becomes hhf (xj)iiE := N a0 n Y i=1 Z 1 0 dxix −1/2 i (1 − xi) d−3/2Y k<l (xk− xl)2f (xj) , (C.3) where N = 2 2(d−1)n+(n−1)(2n−1) πnn! . (C.4)
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The averages we need to compute can be read down from (3.3), and are
I1 := ** n X i=1 cos−2θi 2 ++ E = ** n X i=1 1 1 − xi ++ E , J := ** n X i,j=1 cos−2 θi 2 cos −2θj 2 ++ E = ** n X i,j=1 1 (1 − xi)(1 − xj) ++ E , K := ** n X i=1 cos−4θi 2 ++ E = ** n X i=1 1 (1 − xi)2 ++ E . (C.5)
For notational convenience we gave the names J and K to the corresponding averages,
since they will be found as linear combinations of other quantities named I2 and I3, in
terms of which the SDW coefficients are presented in the paper.
Let us focus now on the factor a0, that gives the degrees of freedom of the model. In
the xi variables it is given by
a0 = N n Y i=1 Z 1 0 dxix −1/2 i (1 − xi)d−3/2 Y k<l (xk− xl)2 . (C.6)
There is a well known result by Selberg [63,64] for such kind of integrals, that gives:
Sn(α, β) := n Y i=1 Z 1 0 dxixαi(1 − xi)β Y k<l (xk− xl)2 = n Y k=1 k!Γ(k + α)Γ(k + β) Γ(k + n + α + β) , (C.7)
from which we obtain, after inserting the factor (C.4) and rearranging the product in (C.7):
a0 = N Sn −12, d − 32 = 2n−1 (2d − 2)! [(d − 1)!]2 n−1 Y k=1 k(2k − 1)! (2k + 2d − 3)! (2k + d − 2)! (2k + d − 1)! , (C.8)
that indeed coincides with the result found in [1].
To proceed further, let us consider the following generalization of Selberg’s integral by
Aomoto [63,65]: Sn,1(α, β; t) := n Y i=1 Z 1 0 dxixαi(1 − xi)β(xi− t) Y k<l (xk− xl)2 = Sn(α, β) n! Y k (k + n + α + β)P (α,β) n (1 − 2t) , (C.9)
where Pn(α,β)(1 − 2t) is the Jacobi polynomial of degree n. By taking a derivative of (C.9)
with respect to t, and evaluating it at t = 1 we get very close to the definition of I1, and
precisely we have I1 = N a0 (−)n∂tSn,1 −12, d − 52; t|t=1= (−)n ∂tSn,1 −12, d − 52; t|t=1 Sn −12, d −32 . (C.10)
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The basic properties of Jacobi polynomials that we need for such calculation are:
dk dzkP (α,β) n (z) = Γ(α + β + n + 1 + k) 2kΓ(α + β + n + 1) P (α+k,β+k) n−k (z) , Pn(α,β)(−1) = (−)nn + β n . (C.11)
We can now compute I1 by inserting (C.9) into (C.10), and using the relations (C.11) and
the result (C.7) we find a quite compact result:
I1= (−)n−1 Sn(−12, d − 52) Sn(−12, d − 32) n!(n + d − 2) Y k (k + n + d − 3) Pn−1(1/2,d−3/2)(1 − 2t)|t=1 = 2n(n + d − 2) 2d − 3 . (C.12)
We now turn to compute the average I2, defined as
I2 := ** X i6=j 1 (1 − xi)(1 − xj) ++ E = J − K . (C.13)
From the definition of Sn,1(α, β; t) in (C.9), it is easy to see that I2 is related to its second
t derivative as I2= N a0 (−)n∂t2Sn,1 −12, d − 52; t|t=1 = (−)n ∂t2Sn,1 −12, d − 52; t Sn −12, d −32 , (C.14)
and in the same way we computed I1 we find for I2
I2 = Sn(−12, d − 52) Sn(−12, d − 32) n!(n + d − 2)(n + d − 1) Y k (k + n + d − 3) P (3/2,d−1/2) n−2 (1 − 2t)|t=1 = 4n(n − 1)(n + d − 1)(n + d − 2) (2d − 1)(2d − 3) . (C.15)
We need at this point to introduce one further generalization of Selberg’s integral,
provided by Kaneko [63]: Kn(α, β; t) := n Y i=1 Z 1 0 dxixαi(1 − xi)β(1 − txi)−1 Y k<l (xk− xl)2 = Sn(α, β)2F1(n, n + α; 2n + α + β; t) , (C.16)
where 2F1(a, b; c; t) is the Gauss hypergeometric function. By taking two derivatives with
respect to t in (C.16) and evaluating at t = 1 one finds an average that is related to K by
linear combinations of I1 and I2. We shall then define I3 as
I3 := N a0 ∂t2Kn −12, d − 12; t|t=1= Sn −12, d −12 Sn −12, d −32 ∂ 2 t 2F1 n, n − 12; 2n + d − 1; t|t=1. (C.17)
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In order to perform the computation we need the following properties of the hypergeometric function: dk dzk 2F1(a, b; c; z) = (a)k(b)k (c)k 2F1(a + k, b + k; c + k; z) , (ak) := Γ(a + k) Γ(a) , 2F1(a, b; c; 1) = Γ(c)Γ(c − a − b) Γ(c − a)Γ(c − b) . (C.18)
Using now (C.18) in (C.17), we can compute I3 that results
I3= Sn −12, d −12 Sn −12, d −32 (n)2 n − 122 (2n + d − 1)2 2F1 n + 2, n + 32; 2n + d + 1; t|t=1 = n(n + 1)(4n 2− 1) (2d − 3)(2d − 5) . (C.19)
By using the definition (C.16) and taking the double derivative with respect to t in t = 1
one finds that K is given as the following linear combination:
K = 1 2I3− 1 2I2+ (n + 1) I1− n(n + 1) 2 , (C.20)
whereas, by means of I2 = J − K, one has
J = 1 2I3+ 1 2I2+ (n + 1) I1− n(n + 1) 2 . (C.21)
This concludes our computations of the modular integrals for even N = 2n. Although
the SDW coefficients can be read off straightforwardly from I1, J and K, we choose to
present them in the paper in terms of I1, I2 and I3, since they have much more compact
expressions.
C.2 Odd N
We turn now to compute the modular integrals required for odd N = 2n + 1. The averages needed will have exactly the same structure as the even N case, the only difference being the form of the modular measure. In particular, the only changes needed will be in the prefactor N and in all the generalized Selberg’s formulas, where the parameter α will switch
everywhere from −12 to +12. The averages in the odd case are explicitly given by
hhf (xj)iiO := N a0 n Y i=1 Z 1 0 dxix1/2i (1 − xi)d−3/2 Y k<l (xk− xl)2f (xj) , (C.22)
where we see that the only difference between (C.22) and (C.3) is the power 1/2 instead of
−1/2, that is the α parameter we used in all the previous computations. In addition, the prefactor now reads
N = 2
2(d−1)+n(2n+2d−3)
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The averages have the same definition as before, being
eI1 := ** n X i=1 cos−2 θi 2 ++ O = ** n X i=1 1 1 − xi ++ O , e J := ** n X i,j=1 cos−2θi 2 cos −2 θj 2 ++ O = ** n X i,j=1 1 (1 − xi)(1 − xj) ++ O , e K := ** n X i=1 cos−4 θi 2 ++ O = ** n X i=1 1 (1 − xi)2 ++ O . (C.24)
Again, we will compute eI2 and eI3 instead of eJ and eK. The degrees of freedom factor a0
now reads a0 = N n Y i=1 Z 1 0 dxix1/2i (1 − xi)d−3/2 Y k<l (xk− xl)2 . (C.25)
Everything goes in the same way as it did with even N , and we easily obtain:
a0 = N Sn 12, d − 32 = 2d−2+n d (2d − 2)! [(d − 1)!]2 n−1 Y k=1 (k + d − 1)(2k + 1)! (2k + 2d − 3)! (2k + d − 1)! (2k + d)! , eI1 = N a0 (−)n∂tSn,1 12, d − 52; t|t=1= 2n(n + d − 1) 2d − 3 , eI2 = N a0 (−)n∂t2Sn,1 12, d −52; t|t=1= 4n(n − 1)(n + d)(n + d − 1) (2d − 1)(2d − 3) , eI3 = N a0 ∂t2Kn 12, d − 12; t|t=1= n(n + 1)(2n + 1)(2n + 3) (2d − 3)(2d − 5) . (C.26)
Also the relations that give eJ and eK remain unchanged and are
e K = 1 2eI3− 1 2eI2+ (n + 1) eI1− n(n + 1) 2 , e J = 1 2eI3+ 1 2eI2+ (n + 1) eI1− n(n + 1) 2 . (C.27) References
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